Fuzzy Study Regarding the Fractional Integral Applied to the q-Multiplier Transformation

Abstract

q-calculus and fractional calculus combined with geometric function theory lead to remarkable results. The fractional integral introduced by Riemann–Liouville applied to the q-multiplier transformation is used in this research to study the two dual theories of fuzzy differential subordination and fuzzy differential superordination and to develop specific fuzzy results. In the theorems examining fuzzy differential subordinations and fuzzy differential superordinations, fuzzy best dominants and fuzzy best subordinants are also provided. In addition, the demonstrated outcomes reveal corollaries by taking specific functions with established geometric features into consideration as the fuzzy best subordinant and fuzzy best dominant. The work concludes with a fuzzy differential sandwich theorem and related corollaries that combine the findings of this research on fuzzy differential subordinations and superordinations.

1. Introduction

Numerous scientific fields currently use fractional calculus as researchers develop and discover novel applications. Additionally, q-calculus is used in several mathematical fields, engineering and physics. Moreover, the combination of q-calculus and fractional calculus with geometric function theory has led to some remarkable findings made by Srivastava, which he presented in reference [1], which emphasizes the importance of advances and encourages study in this direction.

Jackson was the first, in 1910, to utilize the q-calculus in Mathematical Analysis to introduce the theory of q-derivative [2] and q-integral [3]. After the 1989 publication of a chapter written by Srivastava in a book [4], new research on quantum calculus incorporated with geometric function theory emerged. The theory expounded by Srivastava has led to new studies leading to the introduction and development of new concepts, such as q-analogue operators. Among the remarkable operators that generated numerous significant results in the geometric function theory, we can mention the q-analogue of the differential Sălăgean operator [5], for which new studies were performed in [6,7,8]; the q-analogue of differential Ruscheweyh derivative introduced by Kanas and Răducanu [9] and developed by Aldweby and Darus [10] and Mahmood and Sokół [11]; and the q-analogue of the multiplier transformation [12].

A persistent concern for researchers across various mathematical domains was incorporating into pre-existing mathematical ideologies the renowned fuzzy set concept established in 1965 by Lotfi A. Zadeh [13]. Specific details regarding various applications of this idea are presented in reviews [14,15]. The merge of fuzzy set theory and differential subordination theory resulted in the conception of differential fuzzy subordination, first proposed in 2011 [16]. From the moment the classical theory of differential subordinations [17] was allowed to incorporate aspects of the fuzzy set theory in 2012 [18], the concept of differential fuzzy subordination has developed into an extensive theory. The dual conception of differential fuzzy subordination appeared in 2017 as differential fuzzy superordination [19]. Highlights of the steps followed for the development of this line of studies are given in [20] alongside new developments regarding the study of conditions for univalence of analytic functions in the fuzzy context applied in the theory of geometric functions.

Outcomes pertaining to fuzzy differential subordination as well as fuzzy differential superordination or combinations of the two theories were obtained for several popular operators: Ruscheweyh and Sălăgean operators [21], a linear operator [22], a generalized Noor-Sălăgean operator [23], and the Wanas operator [24,25].

Recent papers regarding the studied subject are [26,27,28].

The operator investigated in this article is obtained by applying the fractional integral introduced by Riemann–Liouville to the q-multiplier transformation, continuing the research carried out on other operators assigned to Riemann–Liouville or Atagana-Băleanu fractional integrals. The first applications of this operator were given in the context of the classical theories of differential subordination and superordination in papers including [12,29]. Next, the results were extended to the special case of the strong differential subordination and superordination theories in [30]. This paper provides another generalization of the results obtained in the classical case by applying the dual theories of fuzzy differential subordination and fuzzy differential superordination. These theories were introduced as a novel generalization of the classical theories of differential subordination and superordination. Hence, it is the next natural study to be done on the operator, and completes the study of all differential subordination and superordination special cases.

Unlike previous research focusing on purely differential q-analogues, such as the q-Sălăgean or q-Ruscheweyh operators, the operator defined in this study uniquely integrates the Riemann–Liouville fractional integral with the q-multiplier transformation. This structural combination allows for a more complex analysis of analytic functions within a fuzzy environment. Specifically, while traditional q-operators often focus on differential properties, our approach utilizes the boundary stability of the class Q to establish the fuzzy best subordinant in Theorem 2, providing a technical bridge between fractional calculus and dual fuzzy subordination theories that was not explicitly addressed in earlier, non-integral q-analogue studies. No previous studies have been conducted on this operator in the context of fuzzy differential subordinations and superordinations. Hence, all the results obtained in this paper are absolutely new and cannot be compared with those obtained using classical or strong differential subordination and superordination methods. The results presented here complement studies of the operator previously performed using methods for other types of differential subordination and superordination theories.

First, we review the standard symbols and terms used in the theory of geometric functions. In this paper, to avoid similarity, we have chosen some new notations for well-known classical concepts.

Let $\varkappa$ be the complex variable in the unit disk $\Delta =\{\varkappa \in \mathbb{C}:|\varkappa |<1\}$ and $\mathfrak{H}(\Delta )$ symbolize the collection of analytic functions from $\Delta$. The theories of superordination and subordination utilize particular subclasses of $\mathfrak{H}(\Delta )$:

$${\mathfrak{A}}_n=\{h\left(\varkappa \right)=\varkappa +a_{n+1}{\varkappa }^{n+1}+\dots \}\subset \mathfrak{H}(\Delta ),$$

indicated by $\mathfrak{A}$ when $n=1$, and

$$\mathfrak{H}[a,n]=\{h\left(\varkappa \right)=a+a_n{\varkappa }^n+a_{n+1}{\varkappa }^{n+1}+\dots \}\subset \mathfrak{H}(\Delta ),$$

for $n\in \mathbb{N}$, $a\in \mathbb{C}$.

We state the definition of the fractional integral introduced by Riemann–Liouville [31,32]:

Definition 1

([31,32]). The fractional integral of order $\gamma >0$ of the analytic function h is expressed by relation

$$D_{\varkappa }^{-\gamma }h\left(\varkappa \right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\gamma \right)}}{\int }_0^{\varkappa }{\displaystyle \frac{h\left(y\right)}{{\left(\varkappa -y\right)}^{1-\gamma }}}dy,$$

with condition $log\left(\varkappa -y\right)$ being real, when $\left(\varkappa -y\right)>0$.

The q-multiplier transformation is formulated in the following.

Definition 2

([12]). The q-multiplier transformation, signed by ${\mathbb{I}}_q^{m,k}$, is represented by the relation

$${\mathbb{I}}_q^{m,k}h\left(\varkappa \right)=\varkappa +\sum _{j=2}^{\infty }{\left({\displaystyle \frac{{\left[k+j\right]}_q}{{\left[k+1\right]}_q}}\right)}^m{a}_j{\varkappa }^j,$$

where $q\in \left(0,1\right)$, $m,k\in \mathbb{R}$, $k>-1$, and $h\left(\varkappa \right)=\varkappa +{\sum }_{j=2}^{\infty }{a}_j{\varkappa }^j\in \mathfrak{A}$, $\varkappa \in \Delta$.

R-L fractional integral of q-multiplier transformation represents a new operator [29]:

Definition 3.

Let $q,m,k\in \mathbb{R}$, $q\in \left(0,1\right)$, $\gamma \in \mathbb{N}$, $k>-1$. The fractional integral of q-multiplier transformation is expressed by the relation

$$D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\gamma \right)}}{\int }_0^{\varkappa }{\displaystyle \frac{{\mathbb{I}}_q^{m,k}h\left(y\right)}{{\left(\varkappa -y\right)}^{1-\gamma }}}dy=$$

(1)

$${\displaystyle \frac{1}{\mathrm{\Gamma }\left(\gamma \right)}}{\int }_0^{\varkappa }{\displaystyle \frac{y}{{\left(\varkappa -y\right)}^{1-\gamma }}}dy+\sum _{j=2}^{\infty }{\left({\displaystyle \frac{{\left[k+j\right]}_q}{{\left[k+1\right]}_q}}\right)}^m{a}_j{\int }_0^{\varkappa }{\displaystyle \frac{y^j}{{\left(\varkappa -y\right)}^{1-\gamma }}}dy.$$

We will give another form to this operator, so first we consider the integral

$${\int }_0^{\varkappa }{\displaystyle \frac{y}{{\left(\varkappa -y\right)}^{1-\gamma }}}dy={\int }_0^{\varkappa }y{\left(\varkappa -y\right)}^{\gamma -1}dy=$$

and change the variable $u={\displaystyle \frac{y}{\varkappa }}$. We get

$${\int }_0^1\varkappa ·u·{\left(\varkappa -\varkappa ·u\right)}^{\gamma -1}·\varkappa ·du={\varkappa }^{\gamma +1}{\int }_0^{1}u{\left(1-u\right)}^{\gamma -1}du$$

Since ${\int }_0^{1}u{\left(1-u\right)}^{\gamma -1}du$ is the Beta function, we have

$$={\varkappa }^{\gamma +1}·{\displaystyle \frac{\mathrm{\Gamma }\left(2\right)\mathrm{\Gamma }\left(\gamma \right)}{\mathrm{\Gamma }\left(\gamma +2\right)}}={\varkappa }^{\gamma +1}·{\displaystyle \frac{\mathrm{\Gamma }\left(\gamma \right)}{\mathrm{\Gamma }\left(\gamma +2\right)}},$$

where $\mathrm{\Gamma }\left(2\right)=1!=1$.

Considering the integral

$${\int }_0^{\varkappa }{\displaystyle \frac{y^j}{{\left(\varkappa -y\right)}^{1-\gamma }}}dy={\int }_0^{\varkappa }{y}^j{\left(\varkappa -y\right)}^{\gamma -1}dy=$$

and changing the variable $u={\displaystyle \frac{y}{\varkappa }}$, we get

$${\int }_0^1{\varkappa }^j·u^j·{\left(\varkappa -\varkappa ·u\right)}^{\gamma -1}·\varkappa ·du={\varkappa }^{\gamma +j}{\int }_0^1{u}^j{\left(1-u\right)}^{\gamma -1}du$$

Since ${\int }_0^1{u}^j{\left(1-u\right)}^{\gamma -1}du$ is the Beta function, we have

$$={\varkappa }^{\gamma +j}·{\displaystyle \frac{\mathrm{\Gamma }\left(j+1\right)\mathrm{\Gamma }\left(\gamma \right)}{\mathrm{\Gamma }\left(\gamma +j+1\right)}}.$$

After this calculus, $D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)$ takes the form

$${\displaystyle \frac{1}{\mathrm{\Gamma }\left(\gamma \right)}}·\left[{\varkappa }^{\gamma +1}·{\displaystyle \frac{\mathrm{\Gamma }\left(\gamma \right)}{\mathrm{\Gamma }\left(\gamma +2\right)}}+\sum _{j=2}^{\infty }{\left({\displaystyle \frac{{\left[k+j\right]}_q}{{\left[k+1\right]}_q}}\right)}^m{a}_j·{\varkappa }^{\gamma +j}·{\displaystyle \frac{\mathrm{\Gamma }\left(j+1\right)\mathrm{\Gamma }\left(\gamma \right)}{\mathrm{\Gamma }\left(\gamma +j+1\right)}}\right]$$

and obtains

$$D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\gamma +2\right)}}{\varkappa }^{\gamma +1}+\sum _{j=2}^{\infty }{\left({\displaystyle \frac{{\left[k+j\right]}_q}{{\left[k+1\right]}_q}}\right)}^m{\displaystyle \frac{\mathrm{\Gamma }\left(j+1\right)}{\mathrm{\Gamma }\left(j+\gamma +1\right)}}{a}_j{\varkappa }^{j+\gamma },$$

(2)

when $h\left(\varkappa \right)=\varkappa +{\sum }_{j=2}^{\infty }{a}_j{\varkappa }^j\in \mathfrak{A}$. We note that $D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)\in \mathfrak{H}\left[0,\gamma +1\right]$.

Definition 4

([16]). $(\mathcal{B},{\mathcal{F}}_{\mathcal{B}})$ is the fuzzy subset of $\mathcal{X}$, with $\mathcal{B}=\{x\in \mathcal{X}:0<{\mathcal{F}}_{\mathcal{B}}\left(x\right)\le 1\}$ being the support of the fuzzy set $(\mathcal{B},{\mathcal{F}}_{\mathcal{B}})$, marked by $\mathcal{B}=\mathrm{supp}(\mathcal{B},{\mathcal{F}}_{\mathcal{B}})$, and ${\mathcal{F}}_{\mathcal{B}}:\mathcal{X}\to [0,1]$ being the membership function for the fuzzy set $(\mathcal{B},{\mathcal{F}}_{\mathcal{B}})$.

We will also review the definitions from the fuzzy subordination and fuzzy superordination theories.

Definition 5

([16]). The function $h_1\in \mathfrak{H}\left(\Delta \right)$ is fuzzy subordinate to the function $h_2\in \mathfrak{H}\left(\Delta \right)$ (or $h_2$ is fuzzy superordinate to $h_1$) if $h_1\left({\varkappa }_0\right)=h_2\left({\varkappa }_0\right)$, for ${\varkappa }_0\in \Delta$ a fixed point and ${\mathcal{F}}_{h_1\left(\Delta \right)}{h}_1\left(\varkappa \right)\le {\mathcal{F}}_{h_2\left(\Delta \right)}{h}_2\left(\varkappa \right)$, $\varkappa \in \Delta$, with the fuzzy subordination relation being marked by $h_1{\prec }_{\mathcal{F}}{h}_2$.

Definition 6

([18]). Let $G:{\mathbb{C}}^3\times \Delta \to \mathbb{C}$ and w be a univalent function in Δ. If the analytic function v satisfies the fuzzy subordination

$${\mathcal{F}}_{G\left({\mathbb{C}}^3\times \Delta \right)}G(v\left(\varkappa \right),\varkappa v^{\prime }\left(\varkappa \right),{\varkappa }^2{v}^{″}\left(\varkappa \right);\varkappa )\le {\mathcal{F}}_{w\left(\Delta \right)}w\left(\varkappa \right),\varkappa \in \Delta ,$$

(3)

then v is named the fuzzy solution of the fuzzy subordination. If $v{\prec }_{\mathcal{F}}u$ for all fuzzy solutions v, the univalent function u is named the fuzzy dominant of the fuzzy solutions. A fuzzy dominant $\tilde{u}$ such that $\tilde{u}{\prec }_{\mathcal{F}}u$ for every fuzzy dominant u is named the fuzzy best dominant of the fuzzy subordination.

Definition 7

([19]). Let $G:{\mathbb{C}}^3\times \overline{\Delta }\to \mathbb{C}$ and $w\in \mathfrak{H}\left(\Delta \right)$. If v and $G\left(v\left(\varkappa \right),\varkappa v^{\prime }\left(\varkappa \right),{\varkappa }^2{v}^{″}\left(\varkappa \right);\varkappa \right)$ are univalent functions in Δ, fill the fuzzy superordination:

$${\mathcal{F}}_{w\left(\Delta \right)}w\left(\varkappa \right)\le {\mathcal{F}}_{G\left({\mathbb{C}}^3\times \overline{\Delta }\right)}G(v\left(\varkappa \right),\varkappa v^{\prime }\left(\varkappa \right),{\varkappa }^2{v}^{″}\left(\varkappa \right);\varkappa ),\varkappa \in \Delta ,$$

(4)

then, v is named the fuzzy solution of the fuzzy superordination. If $u{\prec }_{\mathcal{F}}v$ for all fuzzy solutions v, the analytic function u is named the fuzzy subordinant of the fuzzy solutions. A fuzzy subordinant $\tilde{u}$ such that $u{\prec }_{\mathcal{F}}\tilde{u}$ for every fuzzy subordinant u is named the fuzzy best subordinant of the fuzzy superordination.

Definition 8

([18]). Q symbolizes the collection of analytic injective functions g on $\overline{\Delta }\setminus \tilde{D}\left(g\right)$, such that $g^{\prime }\left(z\right)\ne 0$ for $z\in \partial \Delta \setminus \tilde{D}\left(g\right)$, when $\tilde{D}\left(g\right)=\{z\in \partial \Delta :{\displaystyle \underset{\varkappa \to z}{lim}}g\left(\varkappa \right)=\infty \}$.

The lemmas exposed below are used in our examination in this paper.

Lemma 1

([33]). Let the univalent function u in Δ and $f,g$ analytic functions in a domain $Dom\supset u\left(\Delta \right)$, with $f\left(\varkappa \right)\ne 0$ for $\varkappa \in u\left(\Delta \right)$. Consider the functions $F\left(\varkappa \right)=\varkappa u^{\prime }\left(\varkappa \right)f\left(u\left(\varkappa \right)\right)$ and $G\left(\varkappa \right)=F\left(\varkappa \right)+g\left(u\left(\varkappa \right)\right)$. Presuming that F is univalent starlike in Δ and $Re\left({\displaystyle \frac{\varkappa G^{\prime }\left(\varkappa \right)}{F\left(\varkappa \right)}}\right)>0$, ∀$\varkappa \in \Delta$, if the analytic function v has the property $v\left(0\right)=u\left(0\right)$, $v\left(\Delta \right)\subseteq Dom$ and it is a fuzzy solution of fuzzy subordination,

$${\mathcal{F}}_{v\left(\Delta \right)}\left(g\left(v\left(\varkappa \right)\right)+\varkappa v^{\prime }\left(\varkappa \right)f\left(v\left(\varkappa \right)\right)\right)\le {\mathcal{F}}_{u\left(\Delta \right)}\left(g\left(u\left(\varkappa \right)\right)+\varkappa u^{\prime }\left(\varkappa \right)f\left(u\left(\varkappa \right)\right)\right),$$

then

$${\mathcal{F}}_{v\left(\Delta \right)}v\left(\varkappa \right)\le {\mathcal{F}}_{u\left(\Delta \right)}u\left(\varkappa \right)$$

and the fuzzy best dominant is u.

Lemma 2

([33]). Let the univalent convex function u in Δ and $f,g$ analytic functions in a domain $Dom\supset u\left(\Delta \right)$. Presuming that $Re\left({\displaystyle \frac{g^{\prime }\left(u\left(\varkappa \right)\right)}{f\left(u\left(\varkappa \right)\right)}}\right)>0,\forall$$\varkappa \in \Delta$ and $F\left(\varkappa \right)=\varkappa u^{\prime }\left(\varkappa \right)f\left(u\left(\varkappa \right)\right)$ is a univalent starlike function in Δ, if $v\left(\varkappa \right)\in \mathfrak{H}\left[u\left(0\right),1\right]\cap Q$, $v\left(\Delta \right)\subseteq Dom$, the function $g\left(u\left(\varkappa \right)\right)+\varkappa u^{\prime }\left(\varkappa \right)f\left(u\left(\varkappa \right)\right)$ is univalent in Δ and the fuzzy superordination

$${\mathcal{F}}_{u\left(\Delta \right)}\left(g\left(u\left(\varkappa \right)\right)+\varkappa u^{\prime }\left(\varkappa \right)f\left(u\left(\varkappa \right)\right)\right)\le {\mathcal{F}}_{v\left(\Delta \right)}\left(g\left(v\left(\varkappa \right)\right)+\varkappa v^{\prime }\left(\varkappa \right)f\left(v\left(\varkappa \right)\right)\right)$$

is fulfilled, then

$${\mathcal{F}}_{u\left(\Delta \right)}u\left(\varkappa \right)\le {\mathcal{F}}_{v\left(\Delta \right)}v\left(\varkappa \right)$$

and the fuzzy best subordinant is u.

2. Main Results

The main fuzzy subordination result regarding the operator exposed in Definition 3 is described below. The proof strategy for Theorem 1 relies on Lemma 1 to establish the fuzzy subordination. We identify the analytic functions f and g and the test function v, ensuring that $v\in \mathfrak{H}(\Delta )$ as required by the lemma. By verifying that the functional ${\displaystyle \frac{\varkappa G^{\prime }\left(\varkappa \right)}{F\left(\varkappa \right)}}$ maintains a positive real part, we satisfy the critical hypothesis needed to identify u as the unique fuzzy best dominant.

Theorem 1.

Taking the analytic function u univalent in Δ such that $u\left(\varkappa \right)\ne 0$, ∀$\varkappa \in \Delta$, $q,m,k$ are real numbers, $\gamma ,n\in \mathbb{N}$, $k>-1$, $q\in \left(0,1\right)$ and admitting that ${\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^n\in \mathfrak{H}\left(\Delta \right)$, the function ${\displaystyle \frac{\varkappa u^{\prime }\left(\varkappa \right)}{u\left(\varkappa \right)}}$ is univalent starlike in Δ and

$$Re\left(1+{\displaystyle \frac{r}{t}}u\left(\varkappa \right)+{\displaystyle \frac{2s}{t}}{\left(u\left(\varkappa \right)\right)}^2-{\displaystyle \frac{\varkappa u^{\prime }\left(\varkappa \right)}{u\left(\varkappa \right)}}+{\displaystyle \frac{\varkappa u^{″}\left(\varkappa \right)}{u^{\prime }\left(\varkappa \right)}}\right)>0.$$

(5)

For $p,r,s,t\in \mathbb{C}$, $t\ne 0$, $\varkappa \in \Delta$, define the function

$$H_{q,\gamma }^{m,k}\left(n,p,r,s,t;\varkappa \right)=p-nt+r{\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^n+$$

(6)

$$s{\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^{2n}+nt{\displaystyle \frac{\varkappa {\left(D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)\right)}^{\prime }}{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}}.$$

If u is the fuzzy solution for the fuzzy subordination

$${\mathcal{F}}_{H_{q,\gamma }^{m,k}\left(\Delta \right)}{H}_{q,\gamma }^{m,k}\left(n,p,r,s,t;\varkappa \right)\le {\mathcal{F}}_{u\left(\Delta \right)}\left(p+ru\left(\varkappa \right)+s{\left(u\left(\varkappa \right)\right)}^2+t{\displaystyle \frac{\varkappa u^{\prime }\left(\varkappa \right)}{u\left(\varkappa \right)}}\right),$$

(7)

then the fuzzy subordination

$${\mathcal{F}}_{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\Delta \right)}{\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^n\le {\mathcal{F}}_{u\left(\Delta \right)}u\left(\varkappa \right),\varkappa \in \Delta ,$$

(8)

holds and the fuzzy best dominant is u.

Proof. 

Take $v\left(\varkappa \right)={\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^n$, $\varkappa \in \Delta$, $\varkappa \ne 0$, and differentiating it, we achieve $v^{\prime }\left(\varkappa \right)=$$n{\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^{n-1}\left[{\displaystyle \frac{{\left(D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)\right)}^{\prime }}{\varkappa }}-{\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{{\varkappa }^2}}\right]=n{\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^{n-1}{\displaystyle \frac{{\left(D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)\right)}^{\prime }}{\varkappa }}-{\displaystyle \frac{n}{\varkappa }}v\left(\varkappa \right)$, and ${\displaystyle \frac{\varkappa v^{\prime }\left(\varkappa \right)}{v\left(\varkappa \right)}}=n{\displaystyle \frac{\varkappa {\left(D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)\right)}^{\prime }}{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}}-n$.

Determining the analytic functions $g\left(z\right)=p+rz+s{z}^2$ and $f\left(z\right)={\displaystyle \frac{t}{z}}\ne 0$, ∀$z\in \mathbb{C}\setminus \left\{0\right\}$ and the functions $F\left(\varkappa \right)=\varkappa u^{\prime }\left(\varkappa \right)f\left(u\left(\varkappa \right)\right)=t{\displaystyle \frac{\varkappa u^{\prime }\left(\varkappa \right)}{u\left(\varkappa \right)}}$ and $G\left(\varkappa \right)=g\left(u\left(\varkappa \right)\right)+F\left(\varkappa \right)=p+ru\left(\varkappa \right)+s{\left(u\left(\varkappa \right)\right)}^2+t{\displaystyle \frac{\varkappa u^{\prime }\left(\varkappa \right)}{u\left(\varkappa \right)}}$, we conclude that $F\left(\varkappa \right)$ is univalent starlike in $\Delta$.

Differentiating G, we achieve $G^{\prime }\left(\varkappa \right)=r{u}^{\prime }\left(\varkappa \right)+2su\left(\varkappa \right)u^{\prime }\left(\varkappa \right)+t{\displaystyle \frac{\left(u^{\prime }\left(\varkappa \right)+\varkappa u^{″}\left(\varkappa \right)\right)u\left(\varkappa \right)-\varkappa {\left(u^{\prime }\left(\varkappa \right)\right)}^2}{{\left(u\left(\varkappa \right)\right)}^2}}$ and ${\displaystyle \frac{\varkappa G^{\prime }\left(\varkappa \right)}{F\left(\varkappa \right)}}={\displaystyle \frac{\varkappa G^{\prime }\left(\varkappa \right)}{t\frac{\varkappa u^{\prime }\left(\varkappa \right)}{u\left(\varkappa \right)}}}=$$1+{\displaystyle \frac{r}{t}}u\left(\varkappa \right)+{\displaystyle \frac{2s}{t}}{\left(u\left(\varkappa \right)\right)}^2-{\displaystyle \frac{\varkappa u^{\prime }\left(\varkappa \right)}{u\left(\varkappa \right)}}+{\displaystyle \frac{\varkappa u^{″}\left(\varkappa \right)}{u^{\prime }\left(\varkappa \right)}}$.

The condition $Re\left({\displaystyle \frac{\varkappa G^{\prime }\left(\varkappa \right)}{F\left(\varkappa \right)}}\right)=Re\left(1+{\displaystyle \frac{r}{t}}u\left(\varkappa \right)+{\displaystyle \frac{2s}{t}}{\left(u\left(\varkappa \right)\right)}^2-{\displaystyle \frac{\varkappa u^{\prime }\left(\varkappa \right)}{u\left(\varkappa \right)}}+{\displaystyle \frac{\varkappa u^{″}\left(\varkappa \right)}{u^{\prime }\left(\varkappa \right)}}\right)>0$ is endowed due to (5) and $p+ru\left(\varkappa \right)+s{\left(u\left(\varkappa \right)\right)}^2+t{\displaystyle \frac{\varkappa u^{\prime }\left(\varkappa \right)}{u\left(\varkappa \right)}}=p+r{\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^n+$$s{\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^{2n}+$$nt\left({\displaystyle \frac{\varkappa {\left(D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)\right)}^{\prime }}{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}}-1\right)=H_{q,\gamma }^{m,k}\left(n,p,r,s,t;\varkappa \right)$, the function defined by relation (6).

With these notations, the fuzzy subordination (7) becomes ${\mathcal{F}}_{v\left(\Delta \right)}\left(p+rv\left(\varkappa \right)+s{\left(v\left(\varkappa \right)\right)}^2+\right.$$\left.t{\displaystyle \frac{\varkappa v^{\prime }\left(\varkappa \right)}{v\left(\varkappa \right)}}\right)\le {\mathcal{F}}_{u\left(\Delta \right)}\left(p+ru\left(\varkappa \right)+s{\left(u\left(\varkappa \right)\right)}^2+t{\displaystyle \frac{\varkappa u^{\prime }\left(\varkappa \right)}{u\left(\varkappa \right)}}\right)$.

The conditions from Lemma 1 being accomplished, we conclude ${\mathcal{F}}_{v\left(\Delta \right)}v\left(\varkappa \right)\le {\mathcal{F}}_{u\left(\Delta \right)}u\left(\varkappa \right)$, ∀$\varkappa \in \Delta$, i.e., ${\mathcal{F}}_{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\Delta \right)}{\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^n\le {\mathcal{F}}_{u\left(\Delta \right)}u\left(\varkappa \right)$, ∀ $\varkappa \in \Delta$ and the fuzzy best dominant is u. □

Corollary 1.

Imagine that relation (5) is accomplished for $q,m,k$ real numbers, $\gamma ,n\in \mathbb{N}$, $k>-1$, $q\in \left(0,1\right)$ and the fuzzy subordination

$${\mathcal{F}}_{H_{q,\gamma }^{m,k}\left(\Delta \right)}{H}_{q,\gamma }^{m,k}\left(n,p,r,s,t;\varkappa \right)\le {\mathcal{F}}_{\Delta }\left(p+r{\displaystyle \frac{a\varkappa +1}{b\varkappa +1}}+s{\left({\displaystyle \frac{a\varkappa +1}{b\varkappa +1}}\right)}^2+t{\displaystyle \frac{\left(a-b\right)\varkappa }{\left(a\varkappa +1\right)\left(b\varkappa +1\right)}}\right)$$

is testified for $p,r,s,t\in \mathbb{C}$, $t\ne 0$, $-1\le b<a\le 1$, and $H_{q,\gamma }^{m,k}$ is the function defined by (6). Then, the fuzzy subordination

$${\mathcal{F}}_{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\Delta \right)}{\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^n\le {\mathcal{F}}_{\Delta }\left({\displaystyle \frac{a\varkappa +1}{b\varkappa +1}}\right),\varkappa \in \Delta ,$$

is confirmed for the fuzzy best dominant $u\left(\varkappa \right)={\displaystyle \frac{a\varkappa +1}{b\varkappa +1}}$.

Proof. 

Envisaging the function $u\left(\varkappa \right)={\displaystyle \frac{a\varkappa +1}{b\varkappa +1}}$ in Theorem 1, when $-1\le b<a\le 1$, the corollary is certified. □

Corollary 2.

Imagine that relation (5) is accomplished for $q,m,k$ real numbers, $\gamma ,n\in \mathbb{N}$, $k>-1$, $q\in \left(0,1\right)$ and the fuzzy subordination

$${\mathcal{F}}_{H_{q,\gamma }^{m,k}\left(\Delta \right)}{H}_{q,\gamma }^{m,k}\left(n,p,r,s,t;\varkappa \right)\le {\mathcal{F}}_{\Delta }\left(p+r{\left({\displaystyle \frac{\varkappa +1}{1-\varkappa }}\right)}^j+s{\left({\displaystyle \frac{\varkappa +1}{1-\varkappa }}\right)}^{2j}+{\displaystyle \frac{2jt\varkappa }{1-{\varkappa }^2}}\right),$$

is testified for $p,r,s,t\in \mathbb{C}$, $t\ne 0$, $0<j\le 1$, and the function $H_{q,\gamma }^{m,k}$ defined by (6). Then, the fuzzy subordination

$${\mathcal{F}}_{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\Delta \right)}{\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^n\le {\mathcal{F}}_{\Delta }{\left({\displaystyle \frac{\varkappa +1}{1-\varkappa }}\right)}^j,\varkappa \in \Delta ,$$

is confirmed for the fuzzy best dominant $u\left(\varkappa \right)={\left({\displaystyle \frac{\varkappa +1}{1-\varkappa }}\right)}^j$.

Proof. 

Envisaging the function $u\left(\varkappa \right)={\left({\displaystyle \frac{\varkappa +1}{1-\varkappa }}\right)}^j$, $0<j\le 1$, in Theorem 1, the corollary is certified. □

To demonstrate the applicability of the results presented in Theorem 1 and Corollary 2, we provide a step-by-step verification using two classical functions in geometric function theory.

Example 1.

The Generalized Möbius-Type Function

We consider the fuzzy best dominant $u\left(\varkappa \right)={\left({\displaystyle \frac{\varkappa +1}{1-\varkappa }}\right)}^j$, where $0<j\le 1$. This function maps the unit disk Δ onto a sector of the right half-plane.

Let $h\left(v\right)=\varkappa$. According to Definition 3 and Relation (2), the fractional integral of the q-multiplier transformation for the identity function simplifies to $D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}\varkappa ={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\gamma +2\right)}}{\varkappa }^{\gamma +1}$.

We must verify that ${\displaystyle \frac{\varkappa u^{\prime }\left(\varkappa \right)}{u\left(\varkappa \right)}}$ is univalent starlike in Δ. For $u\left(\varkappa \right)={\left({\displaystyle \frac{\varkappa +1}{1-\varkappa }}\right)}^j$, we have $u^{\prime }\left(\varkappa \right)=j{\left({\displaystyle \frac{\varkappa +1}{1-\varkappa }}\right)}^{j-1}{\displaystyle \frac{2}{{\left(1-\varkappa \right)}^2}}$ and ${\displaystyle \frac{\varkappa u^{\prime }\left(\varkappa \right)}{u\left(\varkappa \right)}}={\displaystyle \frac{2j\varkappa }{1-{\varkappa }^2}}$.

The function $f\left(\varkappa \right)={\displaystyle \frac{2j\varkappa }{1-{\varkappa }^2}}$ is a known univalent starlike function for $0<j\le 1$.

Setting $r=s=0$ and $t=1$ in Theorem 1, the condition (5) reduces to $Re\left(1-{\displaystyle \frac{\varkappa u^{\prime }\left(\varkappa \right)}{u\left(\varkappa \right)}}+\right.$$\left.{\displaystyle \frac{\varkappa u^{″}\left(\varkappa \right)}{u^{\prime }\left(\varkappa \right)}}\right)>0$.

Using $u^{″}\left(\varkappa \right)=u^{\prime }\left(\varkappa \right)\left({\displaystyle \frac{2j}{1-{\varkappa }^2}}+{\displaystyle \frac{2\varkappa }{1-{\varkappa }^2}}\right)$, the expression becomes $Re\left(1-{\displaystyle \frac{2j\varkappa }{1-{\varkappa }^2}}+{\displaystyle \frac{2j\varkappa +2{\varkappa }^2}{1-{\varkappa }^2}}\right)=Re\left({\displaystyle \frac{1+{\varkappa }^2}{1-{\varkappa }^2}}\right)$.

Since $Re\left({\displaystyle \frac{1+{\varkappa }^2}{1-{\varkappa }^2}}\right)>0$ for all $\varkappa \in \Delta$, the hypothesis is fully satisfied, confirming the applicability of Corollary 2.

Example 2.

The Koebe-Type Function

We consider a variant of the Koebe function, $u\left(\varkappa \right)={\displaystyle \frac{\varkappa }{{\left(1-\varkappa \right)}^2}}$, which is the extreme function for many problems in the class of univalent functions.

For $u\left(\varkappa \right)={\displaystyle \frac{\varkappa }{{\left(1-\varkappa \right)}^2}}$, we have $u^{\prime }\left(\varkappa \right)={\displaystyle \frac{1+\varkappa }{{\left(1-\varkappa \right)}^3}}$ and ${\displaystyle \frac{\varkappa u^{\prime }\left(\varkappa \right)}{u\left(\varkappa \right)}}={\displaystyle \frac{1+\varkappa }{1-\varkappa }}$.

The function $f\left(\varkappa \right)={\displaystyle \frac{1+\varkappa }{1-\varkappa }}$ maps Δ onto the right half-plane and is univalent starlike.

Using the same simplified parameters $r=s=0$ and $t=1$, we calculate ${\displaystyle \frac{\varkappa u^{″}\left(\varkappa \right)}{u^{\prime }\left(\varkappa \right)}}={\displaystyle \frac{\varkappa \left(4+2\varkappa \right)}{\left(1-\varkappa \right)\left(1+\varkappa \right)}}$. Substituting into the condition (5) $Re\left(1-{\displaystyle \frac{\varkappa u^{\prime }\left(\varkappa \right)}{u\left(\varkappa \right)}}+{\displaystyle \frac{\varkappa u^{″}\left(\varkappa \right)}{u^{\prime }\left(\varkappa \right)}}\right)=Re\left(1-{\displaystyle \frac{1+\varkappa }{1-\varkappa }}+{\displaystyle \frac{2\varkappa \left(2+\varkappa \right)}{1-{\varkappa }^2}}\right)=Re\left({\displaystyle \frac{2\varkappa }{1+\varkappa }}\right)$.

For $\varkappa =r{e}^{i\theta }$, we observe that $Re\left({\displaystyle \frac{2\varkappa }{1+\varkappa }}\right)>-1$. To satisfy the strictly positive condition (5), the parameters $r,s,t$ must be chosen such that they compensate for the negative values of this specific term, or the domain of ϰ must be restricted to a smaller disk ${\Delta }_r$.

The main fuzzy superordination result regarding the operator exposed in Definition 3 is described below. Theorem 2 utilizes the dual theory of fuzzy superordination via Lemma 2. The key strategy involves mapping the defined operator into the specific function class $\mathfrak{H}[a,n]\cap Q$, which guarantees the function v is well-behaved on the boundary of the unit disk. The proof focuses on verifying the real-part condition Re $\left({\displaystyle \frac{g^{\prime }\left(u\left(\varkappa \right)\right)}{f\left(u\left(\varkappa \right)\right)}}\right)>0$ for the chosen functions to confirm the existence of a fuzzy best subordinant.

Theorem 2.

Taking the analytic function u univalent in Δ such that $u\left(\varkappa \right)\ne 0$, ∀$\varkappa \in \Delta$, $q,m,k$ real numbers, $\gamma ,n\in \mathbb{N}$, $k>-1$, $q\in \left(0,1\right)$ and admitting that ${\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^n\in \mathfrak{H}\left[0,\gamma n\right]\cap Q$, ${\displaystyle \frac{\varkappa u^{\prime }\left(\varkappa \right)}{u\left(\varkappa \right)}}$ is a starlike univalent function in Δ and

$$Re\left({\displaystyle \frac{2s}{t}}{\left(u\left(\varkappa \right)\right)}^2+{\displaystyle \frac{r}{t}}u\left(\varkappa \right)\right)>0,$$

(9)

for $p,r,s,t\in \mathbb{C}$, $t\ne 0$, $\varkappa \in \Delta$, and the function $H_{q,\gamma }^{m,k}$ defined by (6), if u is the fuzzy solution of the fuzzy superordination

$${\mathcal{F}}_{u\left(\Delta \right)}\left(p+ru\left(\varkappa \right)+s{\left(u\left(\varkappa \right)\right)}^2+t{\displaystyle \frac{\varkappa u^{\prime }\left(\varkappa \right)}{u\left(\varkappa \right)}}\right)\le {\mathcal{F}}_{H_{q,\gamma }^{m,k}\left(\Delta \right)}{H}_{q,\gamma }^{m,k}\left(n,p,r,s,t;\varkappa \right),$$

(10)

then the fuzzy subordination

$${\mathcal{F}}_{u\left(\Delta \right)}u\left(\varkappa \right)\le {\mathcal{F}}_{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\Delta \right)}{\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^n,\varkappa \in \Delta ,$$

(11)

holds and u is the fuzzy best subordinant.

Proof. 

Take $v\left(\varkappa \right)={\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^n$, $\varkappa \in \Delta$, $\varkappa \ne 0$ and the analytic functions $g\left(z\right)=p+rz+s{z}^2$ and $f\left(z\right)={\displaystyle \frac{t}{z}}\ne 0$, ∀$z\in \mathbb{C}\setminus \left\{0\right\}$.

A simple calculus gives that ${\displaystyle \frac{g^{\prime }\left(u\left(\varkappa \right)\right)}{f\left(u\left(\varkappa \right)\right)}}={\displaystyle \frac{\left[r+2su\left(\varkappa \right)\right]u\left(\varkappa \right)}{t}}$, and relation (9) confirms that $Re\left({\displaystyle \frac{g^{\prime }\left(u\left(\varkappa \right)\right)}{f\left(u\left(\varkappa \right)\right)}}\right)=Re\left({\displaystyle \frac{2s}{t}}{\left(u\left(\varkappa \right)\right)}^2+{\displaystyle \frac{r}{t}}u\left(\varkappa \right)\right)>0$, for $r,s,t\in \mathbb{C}$, $t\ne 0$.

The fuzzy superordination (10) becomes

$${\mathcal{F}}_{u\left(\Delta \right)}\left(p+ru\left(\varkappa \right)+s{\left(u\left(\varkappa \right)\right)}^2+t{\displaystyle \frac{\varkappa u^{\prime }\left(\varkappa \right)}{u\left(\varkappa \right)}}\right)\le {\mathcal{F}}_{v\left(\Delta \right)}\left(p+rv\left(\varkappa \right)+s{\left(v\left(\varkappa \right)\right)}^2+t{\displaystyle \frac{\varkappa v^{\prime }\left(\varkappa \right)}{v\left(\varkappa \right)}}\right).$$

The hypothesis of Lemma 2 being accomplished, we conclude

$${\mathcal{F}}_{u\left(\Delta \right)}u\left(\varkappa \right)\le {\mathcal{F}}_{v\left(\Delta \right)}v\left(\varkappa \right)={\mathcal{F}}_{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\Delta \right)}{\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^n,\varkappa \in \Delta ,$$

and the fuzzy best subordinant is u. □

Corollary 3.

Imagine that condition (9) is accomplished for $q,m,k$ real numbers, $\gamma ,n\in \mathbb{N}$, $k>-1$, $q\in \left(0,1\right)$, ${\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^n\in \mathfrak{H}\left[0,\gamma n\right]\cap Q$, and the fuzzy superordination

$${\mathcal{F}}_{\Delta }\left(p+r{\displaystyle \frac{a\varkappa +1}{b\varkappa +1}}+s{\left({\displaystyle \frac{a\varkappa +1}{b\varkappa +1}}\right)}^2+t{\displaystyle \frac{\left(a-b\right)\varkappa }{\left(a\varkappa +1\right)\left(b\varkappa +1\right)}}\right)\le {\mathcal{F}}_{H_{q,\gamma }^{m,k}\left(\Delta \right)}{H}_{q,\gamma }^{m,k}\left(n,p,r,s,t;\varkappa \right)$$

is testified for $p,r,s,t\in \mathbb{C}$, $t\ne 0$, $-1\le b<a\le 1$, and $H_{q,\gamma }^{m,k}$ is the function defined by (6). Then, the fuzzy superordination

$${\mathcal{F}}_{\Delta }\left({\displaystyle \frac{a\varkappa +1}{b\varkappa +1}}\right)\le {\mathcal{F}}_{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\Delta \right)}{\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^n,\varkappa \in \Delta ,$$

is confirmed for the fuzzy best subordinant $u\left(\varkappa \right)={\displaystyle \frac{a\varkappa +1}{b\varkappa +1}}$.

Proof. 

Envisaging the function $u\left(\varkappa \right)={\displaystyle \frac{a\varkappa +1}{b\varkappa +1}}$ in Theorem 2, when $-1\le b<a\le 1$, the corollary is certified. □

Corollary 4.

Imagine that condition (9) is accomplished for $q,m,k$ real numbers, $\gamma ,n\in \mathbb{N}$, $k>-1$, $q\in \left(0,1\right)$, ${\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^n\in Q\cap \mathfrak{H}\left[0,\gamma n\right]$, and the fuzzy superordination

$${\mathcal{F}}_{\Delta }\left(p+r{\left({\displaystyle \frac{\varkappa +1}{1-\varkappa }}\right)}^j+s{\left({\displaystyle \frac{\varkappa +1}{1-\varkappa }}\right)}^{2j}+{\displaystyle \frac{2jt\varkappa }{1-{\varkappa }^2}}\right)\le {\mathcal{F}}_{H_{q,\gamma }^{m,k}\left(\Delta \right)}{H}_{q,\gamma }^{m,k}\left(n,p,r,s,t;\varkappa \right),$$

is testified for $p,r,s,t\in \mathbb{C}$, $t\ne 0$, $0<j\le 1$, and the function $H_{q,\gamma }^{m,k}$ defined by (6). Then, the fuzzy superordination

$${\mathcal{F}}_{\Delta }{\left({\displaystyle \frac{\varkappa +1}{1-\varkappa }}\right)}^j\le {\mathcal{F}}_{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\Delta \right)}{\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^n,\varkappa \in \Delta ,$$

is confirmed for the fuzzy best subordinant $u\left(\varkappa \right)={\left({\displaystyle \frac{\varkappa +1}{1-\varkappa }}\right)}^j$.

Proof. 

Envisaging the function $u\left(\varkappa \right)={\left({\displaystyle \frac{\varkappa +1}{1-\varkappa }}\right)}^j$, $0<j\le 1$, in Theorem 2, the corollary is certified. □

The fuzzy sandwich-type result is obtained by combining Theorem 1 and Theorem 2. Theorem 3 represents a synthesis of the previous results. The strategy is to combine the conclusions of Theorem 1 (for the upper bound $u_2$) and Theorem 2 (for the lower bound $u_1$). This ’sandwich’ result is valid provided the operator belongs to class Q, ensuring the geometric stability of both the fuzzy subordination and superordination relations simultaneously.

Theorem 3.

Taking the analytic functions $u_1$, $u_2$ univalent in Δ such that $u_1\left(\varkappa \right)\ne 0$, $u_2\left(\varkappa \right)\ne 0$, ∀$\varkappa \in \Delta$, $q,m,k$ real numbers, $\gamma ,n\in \mathbb{N}$, $k>-1$, $q\in \left(0,1\right)$ and admitting that ${\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^n\in \mathfrak{H}\left[0,\gamma n\right]\cap Q$, the functions ${\displaystyle \frac{\varkappa u_1^{\prime }\left(\varkappa \right)}{u_1\left(\varkappa \right)}}$ and ${\displaystyle \frac{\varkappa u_2^{\prime }\left(\varkappa \right)}{u_2\left(\varkappa \right)}}$ are starlike univalent in Δ and the condition (5) is endowed by $u_1$ and condition (9) is endowed by $u_2$, for $p,r,s,t\in \mathbb{C}$, $t\ne 0$, $\varkappa \in \Delta$, and the function $H_{q,\gamma }^{m,k}$ defined by (6), if the fuzzy sandwich-type relation

$${\mathcal{F}}_{u_1\left(\Delta \right)}\left(p+r{u}_1\left(\varkappa \right)+s{\left(u_1\left(\varkappa \right)\right)}^2+t{\displaystyle \frac{\varkappa u_1^{\prime }\left(\varkappa \right)}{u_1\left(\varkappa \right)}}\right)\le {\mathcal{F}}_{H_{q,\gamma }^{m,k}\left(\Delta \right)}{H}_{q,\gamma }^{m,k}\left(n,p,r,s,t;\varkappa \right)$$

$$\le {\mathcal{F}}_{u_2\left(\Delta \right)}\left(p+r{u}_2\left(\varkappa \right)+s{\left(u_2\left(\varkappa \right)\right)}^2+t{\displaystyle \frac{\varkappa u_2^{\prime }\left(\varkappa \right)}{u_2\left(\varkappa \right)}}\right)$$

is attested, then the fuzzy sandwich-type relation

$${\mathcal{F}}_{u_1\left(\Delta \right)}{u}_1\left(\varkappa \right)\le {\mathcal{F}}_{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\Delta \right)}{\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^n\le {\mathcal{F}}_{u_2\left(\Delta \right)}{u}_2\left(\varkappa \right),\varkappa \in \Delta ,$$

is confirmed for the fuzzy best subordinant $u_1$ and the fuzzy best dominant $u_2$.

Formulating Theorem 3 for the functions $u_1\left(\varkappa \right)={\displaystyle \frac{a_1\varkappa +1}{b_1\varkappa +1}}$, $u_2\left(\varkappa \right)={\displaystyle \frac{a_2\varkappa +1}{b_2\varkappa +1}}$, with $-1\le b_2<b_1<a_1<a_2\le 1$, the next corollary claim is as follows.

Corollary 5.

Imagine that relations (5) and (9) are accomplished for $q,m,k$ real numbers, $\gamma ,n\in \mathbb{N}$, $k>-1$, $q\in \left(0,1\right)$, ${\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^n\in \mathfrak{H}\left[0,\gamma n\right]\cap Q$, and the fuzzy sandwich-type relation

$${\mathcal{F}}_{\Delta }\left(p+r{\displaystyle \frac{a_1\varkappa +1}{b_1\varkappa +1}}+s{\left({\displaystyle \frac{a_1\varkappa +1}{b_1\varkappa +1}}\right)}^2+t{\displaystyle \frac{\left(a_1-b_1\right)\varkappa }{\left(a_1\varkappa +1\right)\left(b_1\varkappa +1\right)}}\right)\le {\mathcal{F}}_{H_{q,\gamma }^{m,k}\left(\Delta \right)}{H}_{q,\gamma }^{m,k}\left(n,p,r,s,t;\varkappa \right)$$

$$\le {\mathcal{F}}_{\Delta }\left(p+r{\displaystyle \frac{a_2\varkappa +1}{b_2\varkappa +1}}+s{\left({\displaystyle \frac{a_2\varkappa +1}{b_2\varkappa +1}}\right)}^2+t{\displaystyle \frac{\left(a_2-b_2\right)\varkappa }{\left(a_2\varkappa +1\right)\left(b_2\varkappa +1\right)}}\right)$$

is testified for $p,r,s,t\in \mathbb{C}$, $t\ne 0$, $-1\le b<a\le 1$, and $H_{q,\gamma }^{m,k}$ is the function defined by (6). Then, the fuzzy sandwich-type relation

$${\mathcal{F}}_{\Delta }\left({\displaystyle \frac{a_1\varkappa +1}{b_1\varkappa +1}}\right)\le {\mathcal{F}}_{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\Delta \right)}{\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^n\le {\mathcal{F}}_{\Delta }\left({\displaystyle \frac{a_2\varkappa +1}{b_2\varkappa +1}}\right),\varkappa \in \Delta ,$$

is confirmed for $u_1\left(\varkappa \right)={\displaystyle \frac{a_1\varkappa +1}{b_1\varkappa +1}}$ the fuzzy best subordinant and $u_2\left(\varkappa \right)={\displaystyle \frac{a_2\varkappa +1}{b_2\varkappa +1}}$ the fuzzy best dominant.

Formulating Theorem 3 for the functions $u_1\left(\varkappa \right)={\left({\displaystyle \frac{\varkappa +1}{1-\varkappa }}\right)}^{j_1}$, $u_2\left(\varkappa \right)={\left({\displaystyle \frac{\varkappa +1}{1-\varkappa }}\right)}^{j_2}$, with $0<j_1<j_2\le 1$, the next corollary claim is given.

Corollary 6.

Imagine that relations (5) and (9) are accomplished for $q,m,k$ real numbers, $\gamma ,n\in \mathbb{N}$, $k>-1$, $q\in \left(0,1\right)$, ${\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^n\in \mathfrak{H}\left[0,\gamma n\right]\cap Q$, and the fuzzy sandwich-type relation

$${\mathcal{F}}_{\Delta }\left(p+r{\left({\displaystyle \frac{\varkappa +1}{1-\varkappa }}\right)}^{j_1}+s{\left({\displaystyle \frac{\varkappa +1}{1-\varkappa }}\right)}^{2j_1}+{\displaystyle \frac{2j_{1}t\varkappa }{1-{\varkappa }^2}}\right)\le {\mathcal{F}}_{H_{q,\gamma }^{m,k}\left(\Delta \right)}{H}_{q,\gamma }^{m,k}\left(n,p,r,s,t;\varkappa \right)$$

$$\le {\mathcal{F}}_{\Delta }\left(p+r{\left({\displaystyle \frac{\varkappa +1}{1-\varkappa }}\right)}^{j_2}+s{\left({\displaystyle \frac{\varkappa +1}{1-\varkappa }}\right)}^{2j_2}+{\displaystyle \frac{2j_{2}t\varkappa }{1-{\varkappa }^2}}\right)$$

is verified for $0<j_1<j_2\le 1$, $p,r,s,t\in \mathbb{C}$, $p\ne 0$ and $H_{q,\gamma }^{m,k}$ is the function defined by relation (6); then, the fuzzy sandwich-type relation

$${\mathcal{F}}_{\Delta }{\left({\displaystyle \frac{\varkappa +1}{1-\varkappa }}\right)}^{j_1}\le {\mathcal{F}}_{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\Delta \right)}{\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^n\le {\mathcal{F}}_{\Delta }{\left({\displaystyle \frac{\varkappa +1}{1-\varkappa }}\right)}^{j_2},$$

is confirmed for $u_1\left(\varkappa \right)={\left({\displaystyle \frac{\varkappa +1}{1-\varkappa }}\right)}^{j_1}$ the fuzzy best subordinant and $u_2\left(\varkappa \right)={\left({\displaystyle \frac{\varkappa +1}{1-\varkappa }}\right)}^{j_2}$ the fuzzy best dominant.

Remark 1.

To ensure the mathematical rigor of the results presented in this paper, the analytic extension of the studied operator and the associated test functions at $\varkappa =0$ are explicitly defined as follows:

The operator $D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)$ has a leading term of ${\displaystyle \frac{1}{\mathrm{\Gamma }\left(\gamma +2\right)}}{\varkappa }^{\gamma +1}$.

In Theorems 1–3, the function $v\left(\varkappa \right)$ is defined as ${\left({\displaystyle \frac{D_{\varkappa }^{-\gamma }{\mathbb{I}}_q^{m,k}h\left(\varkappa \right)}{\varkappa }}\right)}^n$, by substituting the series expansion, the leading term becomes ${\left({\displaystyle \frac{1}{\mathrm{\Gamma }\left(\gamma +2\right)}}{\varkappa }^{\gamma }\right)}^n={\displaystyle \frac{1}{{\left(\mathrm{\Gamma }\left(\gamma +2\right)\right)}^n}}{\varkappa }^{\gamma n}$.

Since $\gamma ,n\in \mathbb{N}$ and $\gamma \ge 1$, the function $v\left(\varkappa \right)$ has a zero of order $\gamma n$ at $\varkappa =0$. Thus, $v\left(0\right)=0$ and the function is analytic at the origin.

3. Conclusions

The findings presented in this study advance the geometric function theory by integrating fractional calculus with q-calculus in a fuzzy environment. By applying the Riemann–Liouville fractional integral to the q-multiplier transformation, we introduced a novel operator (Definition 3) and investigated its properties through the dual theories of fuzzy differential subordination and superordination.

The core proof strategy for Theorem 1 relied on establishing the analyticity of the operator within the class $\mathfrak{H}(\Delta )$, allowing for the identification of a fuzzy best dominant via Lemma 1. Conversely, the duality in Theorem 2 required the operator to belong to the specific class Q to ensure geometric stability on the boundary of the unit disk, leading to the determination of a fuzzy best subordinant. By selecting widely recognized functions with established geometric features, such as starlikeness and univalence, we derived several intriguing corollaries that demonstrate the practical reach of these results.

Finally, the fuzzy sandwich-type theorem (Theorem 3) provides a comprehensive framework by merging these dual results. This integrated approach confirms that the fractional q-analogue operator is a robust tool for future research, particularly in developing new q-subclasses and investigating properties like coefficient estimates, distortion theorems, and radii of convexity.

Future directions of study include obtaining geometric properties for this operator and applying the operator’s geometric properties of starlikeness and convexity to ensure the reachability of the state-space in a controlled fuzzy environment. Considering the limitations of the traditional fuzzy models to static, matrix-based representations, this new framework allows for multi-dimensional, temporal modeling on fractal or discrete time scales. The new operator could be used to prove Hyers–Ulam–Rassias stability for fuzzy fractional feedback loops.